Factoring Trinomials with Leading Coefficient 1
Letâ€™s look closely at an example of finding the product of two binomials using the
distributive property:
(x + 3)(x + 4) 
= (x + 3)x + (x + 3)4 
Distributive property 

= x^{2} + 3x + 4x + 12 
Distributive property 

= x^{2} + 7x + 12 

To factor x^{2} + 7x + 12, we need to reverse these steps. First observe that the coef
ficient 7 is the sum of two numbers that have a product of 12. The only numbers that
have a product of 12 and a sum of 7 are 3 and 4. So write 7x as 3x + 4x:
x^{2} + 7x + 12 = x^{2} + 3x + 4x+ 12
Now factor the common factor x out of the first two terms and the common factor 4
out of the last two terms. This method is called factoring by grouping.

Rewrite 7x as 3x + 4x. 

Factor out common factors. 

Factor out the common factor x + 3. 
Example 1
Factoring a trinomial by grouping
Factor each trinomial by grouping.
a) x^{2} + 9x + 18
b) x^{2}  2x  24
Solution
a) We need to find two integers with a product of 18 and a sum of 9. For a product
of 18 we could use 1 and 18, 2 and 9, or 3 and 6. Only 3 and 6 have a sum of 9.
So we replace 9x with 3x 6x and factor by grouping:
x^{2} + 9x + 18 
= x^{2} + 3x + 6x + 18 
Replace 9x by 3x + 6x. 

= (x + 3)x + (x + 3)6 
Factor out common factors. 

= (x + 3)(x + 6) 
Check by using FOIL. 
b) We need to find two integers with a product of 24 and a sum of 2. For a product
of 24 we have 1 and 24, 2 and 12, 3 and 8, or 4 and 6. To get a product of 24
and a sum of 2, we must use 4 and 6:
x^{2}  2x  24 
= x^{2}  6x + 4x  24 
Replace 2x with 6x + 4x. 

= (x  6)x + (x  6)4 
Factor out common factors. 

= (x  6)(x + 4) 
Check by using FOIL. 
The method shown in Example 1 can be shortened greatly. Once we discover
that 3 and 6 have a product of 18 and a sum of 9, we can simply write
x^{2} + 9x + 18 = (x + 3)(x + 6).
Once we discover that 4 and 6 have a product of 24 and a sum of 2, we can
simply write
x^{2}  2x  24 = (x  6)(x + 4).
In the next example we use this shortcut.
Example 2
Factoring ax^{2} + bx + c with a = 1
Factor each quadratic polynomial.
a) x^{2} + 4x + 3
b) x^{2} + 3x  10
c) a^{2}  5a + 6
Solution
a) Two integers with a product of 3 and a sum of 4 are 1 and 3:
x^{2} + 4x + 3 = (x + 1)(x + 3)
Check by using FOIL.
b) Two integers with a product of 10 and a sum of 3 are 5 and 2:
x^{2} + 3x  10 = (x + 5)(x  2)
Check by using FOIL.
c) Two integers with a product of 6 and a sum of 5 are 3 and 2:
a^{2}  5a + 6 = (a  3)(a  2)
Check by using FOIL.
